Bifocal diffractive lenses based on the aperiodic Kolakoski sequence

In this work, we present a new family of Zone Plates (ZPs) designed using the self-generating Kolakoski sequence. The focusing and imaging properties of these aperiodic diffractive lenses coined Kolakoski Zone Plates (KZPs) are extensively studied. It is shown that under monochromatic plane-wave illumination, a KZP produces two main foci of the same intensity along the axial axis. Moreover, one of the corresponding focal lengths is double the other, property correlated with the involved aperiodic sequence. This distinctive optical characteristic is experimentally confirmed. We have also obtained the first images provided by these bifocal new diffractive lenses.

waveguides 45 , and applied mathematics 46 , among others.Here we present the first diffractive lenses based on this formalism and an analytical expression for the transmittance function is derived.The focusing properties of Kolakoski ZPs (KZPs) are studied by computing the intensity distribution along the optical axis and the evolution of the diffraction patterns transversal to the propagation direction.We show that a diffractive lens constructed according to the Kolakoski sequence is intrinsically bifocal.The corresponding foci are located at given axial positions correlated with the involved self-generating aperiodic sequence.This property is experimentally verified obtaining a very good agreement with the theoretical prediction computed numerically.The first experimental images produced by this kind of aperiodic structures as bifocal diffractive lenses are also reported.

The Kolakoski sequence
In Mathematics, the so-called "run-length sequence" of a given sequence is itself the sequence formed by those positive integers that indicate the number of elements of equal consecutive symbols in the sequence.For example, the run-length sequence of ABBABBBAABAAABB is 12132132 because the first A appears once, the next B terms appear twice, the next A term appears once, the next B terms appear 3 times, and so on.
The Kolakoski sequence 41 , which we consider here, is an aperiodic sequence, which is identical to its own runlength sequence.In mathematical terms, this sequence can be generated from a seed K 1 = {1, 2} .The successive elements of the sequence, K S , are obtained from the previous order, K S−1 , by applying the substitution rule to the j-th element of K S−1 in the flowing way: 1 → 1 and 2 → 11 if j is an odd number and 1 → 2 and 2 → 22 if j is an even number.Therefore, , etc.Note that the run-length sequence of K S is K S−1 .For instance, the sequence K 4 = {1, 2, 2, 1, 1, 2, 1} presents 1 time 1, 2 times 2, 2 times 1, 1 time 2, and 1 time 1, so its run-length sequence is The red points in Fig. 1 represent the length L S of the Kolakoski sequence of order S, i.e., the total number of elements of the sequence K S .These numbers grow exponentially, L S = 2, 3, 5, 7, 10, 15, 23, 34... , so have been represented on a logarithmic scale.
By performing a simple linear-logarithmic fitting, a very good approximation for the length of the Kolakoski sequence is L S ≈ 2 • 1.5 S−1 (blue line in Fig. 1).Furthermore, each sequence presents approximately the same number of type "1" and type "2" elements, i.e., 1.5 S−1 elements.On the other hand, if we determine the ratio between the lengths of two consecutive Kolakoski sequences (see Fig. 2), we obtain so the length of the Kolakoski sequence of order S is approximately 50% larger than that corresponding to the previous order S − 1 .This value is equivalent to the golden ratio of the Fibonacci sequence 32 , but in this case, we obtain the rational number ϕ = 3/2 .Therefore, the approximated length of the Kolakoski sequence can be expressed as L S ≈ 2.ϕ S−1 .

Kolakoski zone plate design
Based on the Kolakoski sequences, we can design new aperiodic phase binary ZPs.Each one of these sequences, K S , is used to define the phase transmission generating function φ S (ζ ) , with compact support on the interval ζ ∈ [0, 1] .This interval is partitioned in L S sub-intervals of length d S = 1/L S .The phase transmittance value, φ S,j , that takes at the j-th sub-interval is associated with the element, K S,j , being φ S,j = πK S,j , so φ S,j = π when K S,j is "1" and φ S,j = 2π or, what is the same, φ S,j = 0 when K S,j is "2" (see Fig. 3).
In mathematical terms, the phase transmission function, φ S (ζ ) , can be written as: (1) where ζ = (r/a) 2 is the normalized squared radial coordinate, r is the radial coordinate, a is the lens radius, and "rect" refers to the rectangular function.Figure 4 shows the corresponding phase distribution of a ZP based on the Kolakoski sequence of order S = 7 .Note that the number of concentric annular zones of a KZP of order S coincides with L S .For the case considered in Figs. 3 and 4, the number of zones is L 7 = 23 with approximately the same number of zones with phase π (12 zones) and phase 2π or 0 (11 zones).

Focusing properties
To evaluate the focusing propierties of the Kolakoski lenses, we have computed the axial irradiance provided by these aperiodic zone plates under a monochromatic plane wave illumination, using the Fresnel-Kirchhoff diffraction theory as 47 : where u = a 2 2 z is the reduced axial coordinate, z is the axial distance from the lens plane to the observation plane, and is the wavelength of the incident light.If we consider the phase transmittance function given in equation ( 2), we obtain: (3) 2, 1, 2, 1, 1} .Note that the phase function takes values π or 2π (phase 0) at the j-th sub-intervals of K 7 where K 7,j is "1" or "2", respectively.
where e iπK S,j = (−1) K S,j is the transmittance value that takes the Kolakoski lens of order S at the j-th zone.We have computed the normalized axial irradiance, corresponding to the first diffraction order, provided by KZPs of orders S = 7, 8, and 9.The corresponding numbers of phase zones are 23, 34, and 50 for S = 7, 8, and 9, respectively.As can be seen in Fig. 5, the axial irradiance distributions, represented against the reduce axial coordinate, u, show that the aperiodic ordering of phase zones according to the Kolakoski sequence produces two symmetrical foci around the first diffraction order located at u 1 = L S /2 ≈ ϕ S−1 .Higher diffraction orders also appear due to the binary nature of the structure (not shown in Fig. 5), so these two foci are periodically replicated along the coordinate u with period u p = 2u 1 = L S ≈ 2ϕ S−1 .Note that the resulting main reduced focal lengths u a and u b approximate to L S /3 ≈ 2ϕ S−1 /3 and 2L S /3 ≈ 4ϕ S−1 /3 , so the ratio between the focal distances is u b /u a ≈ 2 .Moreover, the ratios u 1 /u a and u p /u b approximate to the rational number ϕ = 3/2 involved in the Kolakoski aperiodic sequence.The higher the order of the sequence, the better these approximations will be.For example, for S = 9, the irradiance distribution period is u p = 50 , the first diffraction order is located at u 1 = 25 , and the corresponding main focal distances are obtained numerically at u a = 16.802 and u b = 33.198 ,so u b /u a = 1.976 , u 1 /u a = 1.488 , and u p /u b = 1.506.
To contextualize our results within the framework of aperiodic diffractive lenses, the focusing properties of the KZP have been compared with those of the equivalent periodic ZP and other aperiodic intrinsically bifocal ZPs, such as the Fibonacci 32 ZP and the Tribonacci 34 ZP. Figure 6 shows the axial irradiance provided by the first 40 zones of these lenses for comparison.These distributions have been normalized to the maximum intensity  www.nature.com/scientificreports/achieved by the periodic ZP.All these aperiodic lenses split the main focus into a pair of foci with the same axial irradiance, and their separation with respect to the main focal position depends on the properties of the aperiodic sequence.The maximum intensity provided by the KZP is lower compared to the Fibonacci and Tribonacci ZPs, but it also achieves the highest ratio between the focal lengths, with u b /u a ≈ 2 for the KZP, u b /u a ≈ 1.615 for the Fibonacci ZP, and u b /u a ≈ 1.189 for the Tribonacci ZP, providing more options when designing a diffractive lens with specific applications.

Experimental setup
The focusing properties of KZPs were tested on the experimental setup shown in Fig. 7.A collimated and linearly polarized beam from an He-Ne Laser ( = 633 nm) illuminates the liquid crystal spatial light modulator (SLM) (Holoeye PLUTO, 1920 × 1080 pixels, pixel size 8 µ m, 8-bit gray-level) where the designed lenses were implemented.The SLM operates in phase-only modulation mode.A linear phase grating was added to the lens modulation; in this way, the addressed signal is deflected to the first diffraction order in the Fourier plane of the  lens L3.In addition, the SLM is slightly tilted to correct the linear phase and a pinhole (PH) is positioned at the Fourier plane to eliminate all diffraction orders of the linear phase grating except order +1.The PH also prevents noise from the specular reflection (zero diffractive order) and the pixelated structure of the SLM (higher diffraction orders).Then, the SLM plane is imaged through a telescopic system (L2 and L3).In this way, the studied lens transmittance is projected at the exit pupil plane and its focusing binary profile can be captured along the axis by a camera sensor mounted on a motorized stage.
In order to evaluate the imaging properties of this lens, we modified the previous experimental setup, as illustrated in Fig. 7.b.As illumination source, we replaced the He-Ne laser beam by a collimated LED with a chromatic filter, corresponding to = 633 nm, and a binary object with the letters DiOG (Diffractive Optics Group) (see the inset in Fig. 7).

Results
We assessed the focusing properties of a KZP of order S = 8 and radius a = 1.80 mm. Figure 8 shows the experimental axial irradiance distribution along with the one obtained numerically using Eq. ( 3).Both results are in good agreement.It can be seen that the Kolakoski lens provides two foci with very similar intensities whose axial positions are z a = 226.5 mm and z b = 111.7 mm.The corresponding experimental reduced axial coordinates, u = a 2 2 z , are u a = 11.25 and u b = 22.83 , respectively.As predicted from the theoretical analysis, the ratio between the positions of these foci approximates to u b u a = 2.03 ≈ 2 .Moreover, if we compute the ratios u 1 u a and u p u b , where u 1 = 17 and u p = 34 for the Kolakoski sequence of order 8, we can see that they both approximate to ϕ : u 1 u a ≈ 1.5 , To provide a more extensive study of the focusing characteristics of the KZP, the transversal irradiance distribution in the xz plane was also captured experimentally (Fig. 9).This result confirms the bifocal behavior of the lens as well as the corresponding ratio between its focal lengths.Finally, the monochromatic images for the aforementioned wavelength provided by the KZP were captured at several axial positions in the range [88 mm -288 mm] (see Supplementary video).As expected, this ZP produces two focused images of the object at positions 111.7 mm and 226.5 mm where the two foci are located (see Fig. 10).Some halos surrounding the DiOG letters in the first focus can be noticed, since the out-of focus images corresponding to the higher diffraction orders, are superimposed to the in-focus image.On the other

Discussion
A diffractive lens based on the aperiodic Kolakoski sequence has been presented and studied both numerically and experimentally.It was shown that a KZP produces two foci along the optical axis being the corresponding focal lengths correlated with the involved aperiodic Kolakoski sequence.The image-forming capabilities of these bifocal lenses were also tested.We believe that the proposed aperiodic diffractive lens could be of benefit across a broad range of applications where conventional ZPs are currently applied, such as X-ray microscopy, THz imaging, and ophthalmology.Our next step is to design kinoform-type diffractive structures based on this sequence.This step would aim to improve the diffraction efficiency of the lens, thus extending its suitability to an even broader spectrum of optical applications.

Figure 1 .
Figure 1.Length L S of the Kolakoski sequence of order S (red points).The result of the linear-logarithmic fit is also included in the figure (blue line).

2 ,Figure 2 .
Figure 2. Ratio between the lengths of two consecutive Kolakoski sequences, L S /L S−1 .

Figure 6 .
Figure 6.Comparison between the axial irradiances distributions produced by Kolakoski, Fibonacci, Tribonacci and a periodic ZPs.

Figure 7 .
Figure 7. Scheme of the experimental setup used to evaluate (a) the focusing and (b) imaging properties of the KZP.

Figure 8 .
Figure 8. Theoretical and experimental axial irradiance profiles of the Kolakoski lens of order S = 8 .Both of these distributions are normalized with respect to the maximum intensity..

Figure 9 .
Figure 9. Evolution of the transverse intensity distribution produced by a KZP of order 8.